Mathematical Sciences: Vortices in Algebraic and Symplectic Geometry

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This award supports research in the area of vortices in algebraic and symplectic geometry. It is based on earlier work by the investigator on vortices and stable holomorphic pairs. Vortices are minima of Yang-Mills-Higgs energy, and holomorphic pairs are holomorphic vector bundles with prescribed holomorphic sections. This research concerns new directions in the study of vortices and stable pairs and involves a generalization of the notion of a pair to that of a triple consisting of two holomorphic bundles together with a holomorphic bundle homomorphism. This will address the way in which objects can be obtained by a dimensional reduction of an equivariant stable bundle on the Cartesian product of the Riemann surface with the projective line. This award is in the general area of geometry and, in particular, in algebraic and symplectic geometry. This research will aid in our understanding of 4-dimensional manifolds and are central to the current research in 'gauge theory'. In lay terms, a 'manifold' plays the role of a surface.